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Japanese Journal of Applied Physics 55, 011301 (2016) http://doi.org/10.7567/JJAP.55.011301

First-principles calculations of multivacancies in germanium Sholihun1,2, Fumiyuki Ishii1, and Mineo Saito1 1 Division of Mathematical and Physical Science, Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa 920-1192, Japan 2 Department of Physics, Faculty of Mathematics and Natural Science, Universitas Gadjah Mada, Bulaksumur, Yogyakarta 55281, Indonesia

Received May 21, 2015; revised September 29, 2015; accepted October 2, 2015; published online December 4, 2015 We carry out density-functional-theory calculations to study the stability of germanium multivacancies. We use supercells containing 216 atomic sites and simulate two configurations called the “part of hexagonal ring” (PHR) and fourfold configurations of the tri-, tetra-, and pentavacancies. We find that the fourfold configurations of the tetra- and pentavacancies are the most stable and these configurations are also the most stable in the case of silicon. However, we find that the PHR and fourfold configurations have similar energies in the case of the germanium trivacancy. These results are in contrast to those of the silicon trivacancy; the fourfold configuration has substantially lower energy than the PHR configuration. This difference between germanium and silicon is expected to originate from the fact that the four bonds in the fourfold configurations in the germanium trivacancy are weaker than those in the silicon one. By calculating dissociation energies, we find that the silicon tetravacancy is not easy to dissociate, whereas the germanium tetravacancy is not very stable compared with the silicon one. © 2016 The Japan Society of Applied Physics

1.

Introduction

Defects in semiconductors play important roles in the fabrication of semiconductor-based electronic devices, because they have crucial effects on the electronic structures. Recently, germanium has attracted much attention owing to its high carrier mobility.1) Whereas defects in silicon were well studied,2–8) those in germanium are not. Magic numbers have been discussed for Si multivacancies.9) The dangling bond counting (DBC) model, proposed by Chadi and Chang,9) indicates the stability of multivacancies in silicon. According to this model, a decrease in the number of dangling bonds (broken bonds) makes the vacancies more stable. The model leads to so-called magic numbers: the numbers of missing host atoms, n = 4m + 2 (m ¼ 1; 2; 3; . . .), at which vacancies are energetically stable.8–10) Chadi and Chang concluded that the hexavacancy (V6) having a hexagonal ring network of missing atoms and a decavacancy (V10) having a cage network are energetically stable. The stability of the hexavacancy [Fig. 1(a)] was also examined using the density-functional-theory calculations.10–12) It was suggested that the stable configurations of multivacancies in silicon have the “part of hexagonal ring” (PHR) configuration formed by the sequential removal of atoms from a hexagonal ring [Figs. 1(b)–1(d)]. However, Makhov and Lewis13) carried out calculations and found that the fourfold configurations of the tri-, tetra-, and pentavacancies (Figs. 2 and 3) are more stable than the PHR configurations. In the fourfold configurations of tri-, tetra-, and pentavacancies, three, two, and one interstitial atoms are introduced, respectively, into the hexavacancy, and each introduced atom forms four covalent bonds (Figs. 2 and 3).13,14) A very recent deep-level transient spectroscopy (DLTS) measurement has revealed that, in the case of the trivacancy in silicon, the fourfold configuration is energetically more favorable than the PHR configuration,15) which is consistent with the results of Makhov and Lewis. As mentioned above, the stability of silicon multivacancies has been extensively studied. Magic numbers of the multivacancies of GaAs16) and graphene17) were also studied. On the other hand, little is known for germanium multivacancies,

(a)

(b)

(c)

(d)

Fig. 1. (Color online) PHR configurations of the (a) hexa-, (b) tri-, (c) tetra-, and (d) pentavacancies. Red open circles are vacancies.

although germanium mono- and divacancies were well studied by experiments1,18,19) and first-principles calculations.20–24) A classical molecular dynamics (MD) method25) was previously applied to germanium multivacancies but first-principles electronic-structure calculations, which are more reliable than MD calculations, have not been applied to these multivacancies. In particular, to examine the stabilities of fourfold configurations of germanium multivacancies, which were found to be stable in the case of silicon tri-, tetra-, and pentavacancies,13) studies based on first-principles calculations are necessary. In this study, we carry out density-functional-theory calculations to study the stability of germanium vacancies. We also carry out calculations of silicon vacancies and compare the results of germanium and silicon. In the case of the tri-, tetra-, and pentavacancies in silicon, we show that the fourfold configurations have substantially lower formation energy than the PHR ones. The results for germanium tetra-

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Sholihun et al.

(b)

(c)

Fig. 2. (Color online) Fourfold configurations: (a) hexa-, (b) tetra-, and (c) pentavacancies. Red open circles represent vacancies, and blue and black closed circles represent interstitial atoms and their nearest-neighbor atoms, respectively.

the lattice constant by 0.9% whereas the GGA overestimates the lattice constant by 2.6%. In the case of silicon, the LDA underestimates the lattice constant by 0.9% and the GGA overestimates it by 0.6%. We simulate defects by using supercells containing 216 atomic sites. The Brillouin-zone integration is carried out using an 8-k-points Monkhorst–Pack grid (kMP = 8).30) We check the convergence of the k points for the di- and trivacancies by carrying out calculations using kMP = 64 and find that the formation energy varies by 0.07 eV. We also check the convergence of the supercell size. As will be mentioned in Sect. 3, we calculate formation energies of two configurations. When we use the 512-site cell instead of the 216-site cell, the formation energies of the two configurations vary within 0.08 eV. In a previous study, we found that the energy difference between the formation energies of the silicon monovacancy estimated from 216- and 1728-site calculations is 0.06 eV. Therefore, we expect that the present cell (216 sites) is reliable.31) To find the optimized geometry, we fully relax all atoms so that the atomic forces are less than 5 × 10−2 eV=Å. We carry out calculations of the multivacancies for the vacancy size 1 ≤ Nv ≤ 6. We calculate the formation and dissociation energies for each defect system. The formation energy (E f ) is calculated as6,21,32)   M  Nv E f ¼ Ev  ð1Þ EM ; M where EM is the total energy of the perfect supercell consisting of M sites, Ev is the total energy of the vacancy system, and Nv is the number of vacancies. We calculate two types of dissociation energies, D1 and D2, which are, respectively, given by8,10,14,16,17) D1 ¼ E fNv 1 þ E f1  E fNv ; E fNv ,

Fig. 3. (Color online) Fourfold configuration of the trivacancy. The blue closed circles represent the interstitial atom while the black closed circles (A, B, C, and D) represent their four nearest-neighbor atoms.

D2 ¼ E fNv þ1 þ E fNv 1  2E fNv ; E fNv 1 , and E fNv þ1 are the formation

ð2Þ ð3Þ

where energies of the supercell containing Nv, (Nv − 1), and (Nv + 1) vacancies, respectively, whereas E f1 is the formation energy for the monovacancy. 3.

Results and discussion

3.1

and pentavacancies give similar conclusions to those for silicon. However for the trivacancy, we found that the PHR and fourfold configurations have similar formation energies. 2.

Computational methods

We carry out DFT calculations based on the local-density approximation (LDA) for germanium (Ge) and generalized gradient approximation (GGA) for silicon (Si). We use the norm-conserving pseudopotential and cutoff energies of 16 (Ge) and 9 Ry (Si) for the plane wave basis set. The reason why different exchange–correlation functionals are adopted for Si and Ge is that the LDA and GGA well reproduce the experimental lattice constants of germanium and silicon, respectively. The observed lattice constants are 5.65726) and 5.431 Å27) for germanium and silicon, respectively. We use the standard Birch–Murnaghan equation of state28,29) for volume optimization in the calculation of lattice constants. In the case of germanium, the LDA underestimates

Monovacancy We first calculate formation energies of the monovacancy using Eq. (1). For silicon, the calculated formation energy is 3.52 eV, which is in good agreement with the experimental value of 3.6 ± 0.5 eV.7) The value calculated with the 216site cell in the present calculation is close to those calculated with the 1728-site cell in our previous study (3.46 eV).31) The Jahn–Teller effect33) lowers the symmetry, and the optimized geometry has the D2d symmetry. For germanium, we find that the calculated formation energy is 2.32 eV, which is close to the experimental value of 2.35 ± 0.11 eV.34) The determined symmetry is D2d, which is the same as in the case of silicon. However, the germanium monovacancy has a lower formation energy than that of the silicon monovacancy. This lower energy of germanium is expected to be due to the weak covalent bonds. As Fig. 4 shows, the charge density in the bond region in silicon is higher than that in germanium, indicating that the silicon covalent bonds are strong.

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Sholihun et al. Table I. Calculated formation energies (in eV) of the PHR and fourfold configurations in germanium and silicon. Nv is the vacancy size and E f is the formation energy. Nv

E f (Ge) PHR

E f (Si)

Fourfold

PHR

Fourfold

1

2.32

2

4.10

3.52

3 4

5.67 7.14

5.79 6.82

7.08 8.37

6.38 7.16

5

7.96

7.74

8.97

8.23

6

8.49

5.25

9.05

Table II. Calculated bond angles (in deg) of the fourfold configurations in germanium and silicon trivacancies. The atoms denoted by A, B, C, D, and I are shown in Fig. 3. Fig. 4. (Color online) Charge density distributions in the perfect supercell of silicon (top) and germanium (bottom). The unit in the color bar is e=(a.u.)3.

The previous theoretical studies showed that the formation energies of the germanium monovacancy are 2.0–2.9 eV.22–24,35) A recent 64-site calculation based on the Heyd– Scuseria–Ernzerhof range-separated hybrid functional36) gave the formation energy of 2.87 eV for the germanium monovacancy. 3.2

Divacancy We next study the silicon divacancy. On the basis of the results of the electron paramagnetic resonance experiment, the pairing model was proposed37) but a DFT study supported the resonant-bonding (RB) model,38,39) the symmetry of which is the same as that of the pairing model (C2h). We find that the RB configuration has a lower formation energy than the pairing configuration, but the energy difference is very small (0.01 eV). A previous large-scale DFT calculation also revealed that the two configurations have similar energies.38) The 512-site cell calculation indicates that the RB configuration has a 0.02 eV lower formation energy than the pairing configuration. For germanium, we find that the pairing configuration has a slightly lower formation energy (within 0.002 eV) than the RB configuration. Öğüt and Chelikowsky carried out cluster calculations and also found that the two configurations have similar formation energies, i.e., the RB configuration has only 0.03 eV lower formation energy than the pairing configuration.40) We conclude that the RB and pairing configurations have similar energies in both silicon and germanium; thus, the determination of the most stable configurations is still unclear. 3.3

Trivacancy We calculate formation energies of the PHR and fourfold configurations for the trivacancy. In the fourfold configuration, three interstitial atoms are introduced in the hexavacancy (Fig. 3). We find that the geometries of silicon and germanium trivacancies have D3 symmetries. In the case of silicon, we find that the fourfold configuration has a 0.70 eV lower formation energy than the PHR configuration (Table I). On the other hand, we find that the

Bond angles

Germanium

Silicon

∠AIB

163.2

158.4

∠CID

106.2

108.6

∠AIC, ∠BID

86.9

89.6

∠AID, ∠BIC

102.9

103.1

fourfold and PHR configurations have similar formation energies; the fourfold configuration has only a slightly (0.12 eV) higher formation energy than the PHR configuration. When we use the 512-site cell and kMP = 8, the formation energy of the fourfold configuration is larger by 0.08 eV than that of the PHR configuration. Therefore, the difference between the values calculated from the 216- and 512-site cells is only 0.04 eV, so our supercell calculation converges well for the differences between the formation energies of the two configurations. Since the calculated difference between the formation energies of the fourfold and PHR configurations is small, we conclude that the two configurations have similar energies. Each of the interstitial atoms introduced in the hexavacancy forms four bonds (Fig. 3). As shown in Table II, the deviations from the sp3 ideal bond angle (109.5°) for germanium are larger than those for silicon. This result indicates that the four bonds in the germanium trivacancy are weaker than those in the silicon trivacancy. To confirm the weak covalent bonds, we calculate the ratio of the bond-length of each of the four bonds of the interstitial atom to that of the crystal. We find that the ratios in the germanium trivacancy are 1.06–1.13, which are larger than the corresponding values (1.04–1.11) in the silicon trivacancy (Table III). These results also indicate that the bonds in the germanium trivacancy are weak. Moreover, the charge density calculation shows that the four bonds in silicon are stronger than those in germanium, i.e., the bonding charge between the interstitial atom I and its four nearest-neighbor atoms (i.e., atom A and atom C) in germanium is lower than those in silicon (Fig. 5). 3.4

Multivacancies We carry out calculations of the multivacancies with 4 ≤ Nv ≤ 6. In the case of the tetra- and pentavacancies, we carry out calculations of the PHR and fourfold configurations. For silicon, the fourfold configurations have 1.21 and 0.74 eV

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Table III. Calculated bond lengths (L) of the fourfold configurations in germanium and silicon trivacancies. The atoms denoted by A, B, C, D, and I are shown in Fig. 3. The crystal bond lengths (L0) are 2.43 and 2.37 Å for germanium and silicon, respectively. Ratio of the fourfold bond and crystal bond lengths are also shown. Bond

Germanium

Silicon

L (Å)

Ratio

L (Å)

Ratio

I–A

2.74

1.13

2.64

1.11

I–B

2.74

1.13

2.64

1.11

I–C

2.57

1.06

2.47

1.04

I–D

2.57

1.06

2.47

1.04

Fig. 6. Calculated dissociation energies (D1) of the most stable configurations as a function of vacancy size Nv. The dashed and solid lines represent the silicon and germanium cases, respectively.

Fig. 5. (Color online) Charge density distribution in the fourfold configuration of the silicon (top) and germanium (bottom) trivacancies. Atoms denoted by A, C, and I are shown in Fig. 3. The unit in the color bar is e=(a.u.)3.

lower formation energies than do the PHR configurations for the tetra- and pentavacancies, respectively (Table I). The determined symmetries of the fourfold configurations are C2h and C2, respectively. These results are consistent with those of the previous DFT study.13) For germanium, the fourfold configurations of the tetraand pentavacancies have lower formation energies than the PHR configurations, similarly to the case of silicon. However, the formation energy differences are 0.32 and 0.22 eV, respectively, and thus, the differences are smaller than the corresponding values in silicon (Table I). These small formation-energy differences in the germanium tetra- and pentavacancies are expected to be due to the fact that the four bonds of the interstitial atoms are weak, as in the case of the trivacancy. We find the symmetries are C2h and C2 for the tetra- and pentavacancies, respectively, as in the case of silicon. We finally calculate the formation energies of the hexavacancies. The formation energies are 8.49 and 9.05 eV in germanium and silicon, respectively. 3.5

Dissociation energy We here calculate dissociation energies D1 [Eq. (2)] and D2 [Eq. (3)] of the most stable configurations for each Nv-vacancy. D1 corresponds to the dissociation reaction VNv ! VNv 1 þ V, whereas D2 is the dissociation energy for the reaction 2VNv ! VNv þ1 þ VNv 1 . We first calculate D2 of di- to pentavacancies of silicon and find that only the tetravacancy has a positive value. Since D2 corresponds to the second derivative of the formation energy over the vacancy size, Nv, the positive sign of D2 means that

Fig. 7. Calculated dissociation energies (D2) of the most stable configurations as a function of vacancy size Nv. The dashed and solid lines represent the silicon and germanium cases, respectively.

the vacancy is difficult to dissociate. The value of D1 of the tetravacancy is high (Fig. 6), which also indicates that the tetravacancy is stable. From the calculated results of D1 and D2, we conclude that the tetravacancy is stable. We next calculate D2 of the germanium multivacancies (Fig. 7) and find that the signs are all negative from di- to pentavacancies. Therefore, we predict that all these multivacancies are easy to dissociate. Furthermore, the germanium tetravacancy does not have a high value of D1 compared with other multivacancies. Therefore, we expect that the germanium tetravacancy is not very stable compared with the silicon tetravacancy. The stability of the silicon hexavacancy has been confirmed in previous studies,8–10,16) i.e., the sign of D2 in the silicon hexavacancy is positive and D1 is large. Our calculated dissociation energy D1 (Fig. 6) of the germanium hexavacancy is also higher than that of the pentavacancy, implying that the germanium hexavacancy is stable. To further confirm the stability of the germanium hexavacancy, the calculation of the heptavacancy will be necessary to

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examine whether D2 of the germanium hexavacancy is positive, but such a calculation is beyond the scope of this paper. 4.

Conclusions

We carried out density-functional-theory calculations on germanium mono- to hexavacancies. We found prominent differences between the stabilities of germanium and silicon multivacancies. Whereas the fourfold configuration is the most stable in the silicon trivacancy, the PHR and fourfold configurations have similar formation energies in the germanium trivacancy. Although the fourfold configurations are the most stable in the germanium tetra- and pentavacancies, as in the case of silicon ones, the energy differences between the fourfold and PHR configurations are small compared with those in silicon. Therefore, the fourfold configurations of germanium tri-, tetra-, and pentavacancies are not very stable compared with silicon ones, probably because the bonds in the fourfold configurations in germanium multivacancies are weak compared with those in silicon ones. We also elucidated the difference between the dissociation energies of germanium and silicon tetravacancies; by analyzing the calculated dissociation energies, we concluded that the silicon tetravacancy is difficult to dissociate whereas the germanium tetravacancy is not very stable compared with the silicon one. In this study, we found prominent differences between the silicon and germanium multivacancies. Experiments of germanium multivacancies, such as DLTS and electron paramagnetic resonance, are expected to confirm these differences. In particular, we found that PHR and fourfold configurations have similar energies, so one of the two configurations or both are expected to be experimentally observed. The present study focused on mono- to hexavacancies. It will be necessary to clarify the difference between larger multivacancies of germanium and silicon to realize the control of multivacancies in germanium semiconductors. Acknowledgments

This research was partially funded by the MEXT HPCI Strategic Program. This work was partly supported by Grants-in-Aid for Scientific Research (Nos. 25390008, 25790007, 25104714, 26108708, and 15H01015) from the Japan Society for the Promotion of Science (JSPS). The computations in this research were performed using the supercomputers at the Institute for Solid State Physics (ISSP) at the University of Tokyo.

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